Faraday’s Law

FIELDS OF STUDY: Electromagnetism

ABSTRACT: Electricity and magnetism are directly related through Faraday’s law of magnetic induction. Along with Lenz’s, Ampère’s, and Gauss’s laws, Faraday’s law is expressed as one of Maxwell’s equations. It is fundamental to how electrical devices function.

PRINCIPAL TERMS

• electromagnetic field: a region of space in which an electrically charged mass experiences an applied force.
• electromotive force: the energy supplied by an electric source such as a battery, measured in volts.
• Lenz’s law: the principle that the current induced in a conductor by a moving magnetic field flows in the direction opposite to the motion of the magnetic field.
• magnetic flux: the measure of the strength of the magnetic field surrounding an active electrical conductor or a magnet, equal to the product of the external magnetic field (B), the area of the affected surface (A), and the cosine of the angle between them (cosθ).
• magnetic induction: the generation of magnetism within a magnetizable material by proximity to a magnetic field.
• Maxwell’s equations: the mathematical relationships governing electromagnetism, as formulated by James Clerk Maxwell.
• voltage: the difference in electric potential between two points, measured in volts.

Magnets and Electromagnetism

Electricity and magnetism are familiar to most people, but the way in which they are related is not. In physics, both electricity and magnetism are aspects of electromagnetism and are characterized by the existence of an electromagnetic field. Such a field can be a magnetic field, an electric field, or a combination of both. An electric field exists because of electric charge, which can be either positive or negative. Negative electric charge is associated with the electron, while positive charge is associated with the proton.

Magnetism, meanwhile, has been known since ancient times, as it occurs naturally in a type of iron ore. Seafarers used slivers of such ore to construct a simple compass called a lodestone, which always pointed north, just as modern compasses do. Earth’s molten nickel-iron core generates a magnetic field around the planet, and the magnetic needle indicator of a compass aligns with this field such that one end always points toward Earth’s magnetic north pole. The degree to which a compass needle aligns to a magnetic field depends on the magnetic flux within the field. If the magnetic flux is stronger than that of Earth’s field, the compass needle will point toward the stronger magnet. Magnetism can be produced by magnetic induction in any object made of a magnetizable material. For instance, one can pick up an iron nail with a magnet and then use that nail to pick up other nails.

If an electric conductor is moving relative to a magnetic field, the magnetic field will generate an electric current within the conductor. This current will then generate another magnetic field about the conductor. Lenz’s law states that the direction of this induced magnetic field opposes that of the magnetic field that produced the current, creating a repulsion between the two fields. The electric current that flows in the conductor is defined as the movement of electrons from atom to atom. One electron carries a single electrical charge, and so it experiences a force when in a moving magnetic field, as do all electrically charged bodies. Similarly, a magnet in a moving electric field experiences a force acting upon it. This principle, described by Faraday’s law, is the basis for how generators and electric motors function.

Law of Induction

English physicist and chemist Michael Faraday (1791–1867) studied the phenomena of electricity and magnetism. In 1821, he demonstrated that the magnetic field produced by an electric current can induce movement of another magnet. This is the basic principle of the electric motor. Ten years later, in 1831, Faraday showed that a moving magnet induces an electric current in a conductor. He realized that a specific relationship exists between electricity and magnetism, which he described using his law of induction.

Faraday’s law of induction is based on three principles. First, a changing magnetic field induces an electromotive force, or emf, within a conductor. (The abbreviation "emf" is typically lowercased to distinguish it from EMF, for electromagnetic field.) This is simply demonstrated by passing a magnet across a conductor and measuring the resulting voltage between the ends of the conductor. Second, the magnitude of the emf is proportional to the rate at which the magnetic field changes. That is, the faster the magnet is moved past the conductor, the greater the emf produced. Third, the direction of the induced emf depends on the direction in which the magnet moves. This reflects Lenz’s law.---

Simply stated, Faraday’s law says that when a magnetic field moves relative to a conductor, or vice versa, a voltage is induced in the conductor. The magnitude of the voltage produced when the conductor is in motion, called the motional emf, depends directly on the relative velocity of the magnetic field and the conductor. Slow movement produces a low voltage, and fast movement produces a higher voltage. Motional emf is calculated as the product of the magnetic field (B), the length of the conductor (l), and the velocity (v).

Faraday also theorized that there are "magnetic lines of force"—now called magnetic field lines—that make up the magnetic flux. Accordingly, each time a line crosses a conductor, it induces the corresponding emf. The more times that a line can cross the same conductor, the more times that it can induce the emf. Thus, magnetic induction is more effective for a coil of wire than for a straight wire. Each loop of wire within the coil adds to the number of times any magnetic field line can cross that conductor and induce an emf. The total voltage produced is therefore be related directly to the number of "turns" in the coil. The relative speed of the conductor and the magnetic field is expressed as the continuous rate of change of the magnetic flux. Faraday’s law is written as

ξ = −N(Δφ/Δt)

where ξ is the induced emf, measured in volts; N is the number of turns in the coil; and Δφ/Δt is the change in magnetic flux (φ) over time (t), measured in webers (Wb). One weber is equal to one volt-second (V·s). The negative sign shows that the induced emf will oppose the change in flux, per Lenz’s law.

Sample Problem

A coil has 250 turns of wire on it. The magnetic flux passing through it changes from 0.5 webers (Wb) to 4.6 Wb over a period of 3 seconds. What voltage is induced in the coil? Calculate using Faraday’s law.

Answer:

First, to determine the rate of change of the magnetic flux over time (Δφ/Δt), subtract the final flux from the initial flux. Next, divide by the change in time (the final time minus the starting time). In this case, the start time is zero, so the change in time is equal to the final time.

Δφ/Δt = (4.6 Wb − 0.5 Wb) / 3 s

Δφ/Δt = 4.1 Wb / 3 s = 1.367 Wb/s

Next, plug this value into the equation and calculate the induced voltage:

ξ = −N(Δφ/Δt)

ξ = −(250)(1.367 Wb/s)

ξ = −341.75 Wb/s = −341.75 V·s/s

ξ = −341.75 V

The induced voltage is −341.75 volts.

Maxwell’s Equations

As Faraday and other scientists began to understand electromagnetism better through their observations and experiments, the essential question of how electricity and magnetism are transmitted through space was considered. Scottish physicist James Clerk Maxwell (1831–79) theorized that they must travel as electromagnetic waves. On this principle, he consolidated Ampère’s law of magnetism, Gauss’s laws of electricity and magnetism, and Faraday’s law of magnetic induction into precise, related mathematical descriptions. These descriptions are known as Maxwell’s equations.

The relationships described by Maxwell’s equations are central to circuits, almost all of which contain some combination of transformers and inductors. A typical electronic device is the RLC circuit, consisting of a resistor (R), an inductor (L), and a capacitor (C). When designing such circuits, precise values must be calculated for the induced emf of an inductor so that the circuit operates on the input electromagnetic signal as it should. Transformers particularly depend on the principles of magnetic induction for their operation. A transformer consists of two separate coils wrapped around a single iron core and connected to each other through a magnetic field. The output coil of the transformer increases or decreases the voltage supplied by the input coil, depending on the number of turns of wire on each coil.

Bibliography

Calle, Carlos I. Superstrings and Other Things: A Guide to Physics. 2nd ed. Boca Raton: CRC, 2010. Print.

Fitzpatrick, Richard. Maxwell’s Equations and the Principles of Electromagnetism. Hingham: Infinity Science, 2008. Print.

Fleisch, Daniel. A Student’s Guide to Maxwell’s Equations. New York: Cambridge UP, 2008. Print.

Gross, Charles A. Electric Machines. Boca Raton: CRC, 2007. Print.

Kelly, P. F. Electricity and Magnetism. Boca Raton: CRC, 2015. Print.

Nave, R. "Faraday’s Law." HyperPhysics. Georgia State U, n.d. Web. 13 Aug. 2015.

Robbins, Allan H., and Wilhelm C. Miller. Circuit Analysis: Theory and Practice. 5th ed. Clifton Park: Delmar, 2013. Print.